That would depend on the drug you wish to inject. Injecting a vasocompressor or an embolic XXXXXXX into and artery can cause complications. Further, if you try to inject directly in to your Ulnar artery, there is a risk of nerve damage as it runs close to the artery.

However I encourage you against self medication and the option of probing your artery. Rather, talk to your doctor about it. He/she can rule out the possibility of nerve compression and treat you accordingly.